#### Vol. 4, No. 4, 2010

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Algebraic properties of generic tropical varieties

### Tim Römer and Kirsten Schmitz

Vol. 4 (2010), No. 4, 465–491
##### Abstract

We show that the algebraic invariants multiplicity and depth of the quotient ring $S∕I$ of a polynomial ring $S$ and a graded ideal $I\subset S$ are closely connected to the fan structure of the generic tropical variety of $I$ in the constant coefficient case. Generically the multiplicity of $S∕I$ is shown to correspond directly to a natural definition of multiplicity of cones of tropical varieties. Moreover, we can recover information on the depth of $S∕I$ from the fan structure of the generic tropical variety of $I$ if the depth is known to be greater than $0$. In particular, in this case we can see if $S∕I$ is Cohen–Macaulay or almost-Cohen–Macaulay from the generic tropical variety of $I$.

##### Keywords
tropical variety, constant coefficient case, Gröbner fan, generic initial ideals, Cohen–Macaulay, multiplicity, depth
##### Mathematical Subject Classification 2000
Primary: 13F20
Secondary: 14Q99, 13P10
##### Milestones
Received: 11 September 2009
Revised: 5 February 2010
Accepted: 6 April 2010
Published: 13 June 2010
##### Authors
 Tim Römer Institut für Mathematik Universität Osnabrück 49069 Osnabrück Germany Kirsten Schmitz Institut für Mathematik Universität Osnabrück 49069 Osnabrück Germany